Finding Uncertainty in $\theta$ with Fixed $\lambda$

[Hypnotoad]

I'm trying to find the uncertainty in $$\theta$$ where $$\theta$$ is given by:

$$\theta=sin^{-1}\frac{n\lambda}{d}$$

in this case, I am assuming there is no uncertainty in $$\lambda$$.

This is what I tried:

$$\delta \theta=\sqrt{(\frac{d\theta}{dd})^2(\delta d)^2}$$

(the total derivative in there should be a partial derivative, but I don't know how to get that symbol)

$$\delta \theta=\sqrt{(\frac{\frac{\lambda}{d}}{\sqrt{d^2-\lambda^2}})^2\delta d^2}$$

I think that is right, but if I use the values $$\lambda=632.8 nm, d=1.08 \mu m$$ and $$\delta d =.001 \mu m$$ I get an uncertainty of almost 450 degrees. Where am I making my mistake?

 

[dextercioby]

1.Click on this:$$\partial$$.
2.U assumed "n=1",else that derivative should have included the product $n\lambda$.
3.If $n\neq 1$,then you should include "n" in the derivative (under the square root) and redo your calculations.
4.If "n=1",then it's either the numbers are badly chosed,or u ****ed those calculations.

Daniel.

 

[Hypnotoad]

1. Thanks for the latex lesson
2. n=1
3. Turns out it was just my math. Although I did that calculation at least 4 or 5 times and kept getting the same answer before. I don't know what I was doing wrong, but it works much better now.